- •Preface
- •Contents
- •1 Introduction
- •References
- •2.1…Review of Dynamic Engineering Theories of Thin-Walled Beams of Open Section
- •References
- •3.1…Theory of Thin-Walled Beams Based on 3D Equations of the Theory of Elasticity
- •3.1.1 Problem Formulation and Governing Equations
- •3.1.2.1 Solution on the Quasi-Longitudinal Wave
- •3.1.2.2 Solution on the Quasi-Transverse Shear Wave
- •3.2…Construction of the Desired Wave Fields in Terms of the Ray Series
- •References
- •4.2.3 Numerical Example
- •Appendix
- •References
- •5 Conclusion
- •6.3…The Main Kinematic and Dynamic Characteristics of the Wave Surface
- •Reference
6.2 Conditions of Compatibility for Engineering Theories |
83 |
To prove the validity of (2.1), let us multiply (6.6) by G, differentiate over z, and apply formula (6.5). As a result we obtain
Gno |
Z |
|
1 mþ1 |
|
ðn 1Þ! |
d |
|
|
Z;ðmþ1Þ |
|
|||||||||
|
n |
|
Xn 1 |
|
|
|
|
|
|
|
|
|
n 1 |
|
|
m |
|
|
|
|
|
¼ m 0ð Þ |
|
|
|
|
|
|
|||||||||||
|
ozn |
m!ðn 1 mÞ! |
dtn 1 m |
|
|||||||||||||||
|
|
|
¼ |
1 m |
ðn 1Þ! |
d |
|
Z;ðmÞ |
: |
6:7 |
|||||||||
|
|
|
|
|
|||||||||||||||
|
|
|
Xn 1 |
|
|
|
|
|
|
|
|
|
n m |
|
|
|
|
|
|
|
|
|
þ |
ð Þ |
|
|
|
|
|
dtn m |
|
ð Þ |
|||||||
|
|
|
m! n |
|
1 |
m ! |
|
||||||||||||
|
|
|
m¼0 |
|
ð |
|
|
Þ |
|
|
|
|
|
|
|
|
|
In the first sum of (6.7), we substitute m þ 1 by m, in so doing its low limit becomes equal to unit, while the upper limit is equal to n.
Let us separate out the term at m ¼ n in the newly obtained sum and the term at m ¼ 0 in the second sum of (6.7), and add together all remained sums. As a result, we obtain
|
nonZ |
|
|
|
n |
|
|
|
|
dnZ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
G |
|
|
¼ ð 1Þ |
Z;ðnÞ |
|
þ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
|
ozn |
|
dtn |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
n |
m |
|
|
||||||||||||
|
|
|
|
|
|
Xn 1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
|
|
|
|
|
|
1 |
Þ |
m |
|
|
|
ðn 1Þ! |
1 ! þ |
|
ðn 1Þ! |
|
|
d |
|
|
Z;ðmÞ |
; |
|||||||||||
|
|
|
|
|
|
|
|
|
n |
|
|
|
|
n |
|
!m! |
|
dtn m |
||||||||||||||||
|
|
|
|
|
þ ð |
ð |
|
m ! m |
|
|
1 |
|
m |
|
|
|||||||||||||||||||
|
|
|
|
|
|
m¼1 |
|
|
|
|
Þ ð |
|
Þ ð |
|
|
Þ |
|
|
|
|
|
|
|
|
||||||||||
or |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
o |
Z |
|
n |
|
|
|
|
|
|
d |
Z |
|
|
m |
|
|
n! |
|
|
d |
m |
Z; m |
|
|
|
|||||
|
|
|
|
n |
|
|
|
|
|
|
|
|
|
|
|
n |
|
|
Xn 1 |
|
|
|
|
|
|
n |
ð |
|
Þ |
|
|
|||
|
|
n |
|
¼ ð 1Þ |
|
Z;ðnÞ þ |
|
|
|
þ m 1ð 1Þ m!ðn mÞ! |
|
|
|
|
|
|
||||||||||||||||||
|
|
G |
|
|
|
|
|
|
|
: |
ð6:8Þ |
|||||||||||||||||||||||
|
|
|
ozn |
|
dtn |
dtn m |
|
|
||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
¼ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
If we include the second term standing in the right-hand side of (6.8) into the sum, and express the value ð 1ÞnZ;ðnÞ; then we are led to relationship (2.1).
6.3The Main Kinematic and Dynamic Characteristics of the Wave Surface
Now choosing the function ui as the function f in formula (6.4) we have |
|
|
||||||||
u |
G 1 |
½ |
v |
i&kj þ |
oðuikjÞ |
oðuisjÞ |
; |
6:9 |
Þ |
|
½ |
i;j& ¼ |
|
ox |
þ |
oy |
ð |
||||
where ui are the displacement |
vector |
components, |
½ui;j& ¼ ½oui=oxj&; xj |
are the |
spatial rectangular Cartesian coordinates, x- and y- are the main axes of the cross section of the beam (Fig. 3.1), ½ui;ðkÞ& ¼ ½okui=otk&; t is the time, vi ¼ ui;ð1Þ; ki; ki; and si are the components of the unit vectors of the tangential to the centroid axis, and directed along the main axes, respectively, and Latin indices take on the values 1, 2, 3.
84 |
6 Peculiarities of Transient Wave Propagation |
Writing the Hook’s law for a three-dimensional medium in terms of discontinuities and using the condition of compatibility (6.9), we find
½ |
rij |
& ¼ |
G 1k vk dij |
|
G 1l |
vi |
kj |
þ ½ |
vj |
ki |
þ |
k |
||||||
|
|
|
½ & |
|
½ |
& |
|
& |
|
|
||||||||
|
|
|
|
|
oðuikjÞ |
|
|
|
|
|
|
|
|
|
|
|
||
|
|
þ l |
|
þ |
|
oðujkiÞ |
|
þ |
|
oðuisjÞ |
þ |
|||||||
|
|
|
ox |
|
ox |
|
|
oy |
|
|
½ux; x& þ ½ y; y& dij |
|
|
|
||
|
|
u |
|
|
|
|
oðujsiÞ |
; |
ð |
6:10 |
Þ |
|
oy |
||||
|
|
|
where ½vk& ¼ ½vi&ki; |
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
½ |
u |
x; x& ¼ |
|
oðuikiÞ |
¼ |
|
oux |
; |
u |
|
oðuisiÞ |
¼ |
|
ouy |
¼ ½ey& |
; |
|
ox |
|
ox |
oy |
|
oy |
||||||||||
|
|
|
¼ ½ex& ½ |
y; y& ¼ |
|
|
|
k and l are Lame constants, and dij is the Kroneker’s symbol.
Multiplying relationship (6.10) from the right and from the left by kikj and sisj and considering equations
½rxx& ¼ ½rij&kikj ¼ 0; ½ryy& ¼ ½rij&sisj ¼ 0;
what corresponds to the assumption that the normal stresses on the cross-sections parallel to the middle surface could be neglected with respect to other stresses, we obtain
|
|
|
|
|
|
|
½rxx& ¼ G 1k½vk& þ k ½ux; x& þ ½uy; y& þ 2l½ux; x& ¼ 0; |
|
|||
|
|
|
|
|
|
|
½ryy& ¼ G 1k½vk& þ k ½ux; x& þ ½uy; y& þ 2l½uy; y& ¼ 0; |
|
|||
whence it follows that |
|
|
|
||
|
|
½ux; x& ¼ ½uy; y& ¼ |
k |
|
|
|
|
|
½vk&; |
|
|
|
|
2Gðk þ lÞ |
|
||
or |
|
|
|
|
|
|
|
½ux; x& ¼ ½uy; y& ¼ mG 1½vk&; |
ð6:11Þ |
||
since m ¼ |
k |
is the Poisson’s ratio. |
|
|
|
2ðkþlÞ |
|
|
|
Multiplying relationship (6.10) from the right and from the left by kikj; we are led to the equation
½rkk& ¼ ½rij&kikj ¼ G 1ðk þ 2lÞ½vk& þ 2k½ux; x&: |
ð6:12Þ |
|
Substituting (6.11) in (6.12) and considering that E ¼ |
ð3kþ2lÞl |
yields |
kþl |
||
½rkk& ¼ G 1E½vk&; |
|
ð6:13Þ |
where E is the elastic modulus.
Alternatively, multiplying the dynamic condition of compatibility, i.e. the equations of motion rewritten in jumps,
6.3 The Main Kinematic and Dynamic Characteristics of the Wave Surface |
85 |
½rij&kj ¼ qG½vi&; |
ð6:14Þ |
by ki, we obtain |
|
½rkk& ¼ qG½vk&; |
ð6:15Þ |
where q is the density of the beam’s material.
Eliminating the value ½rkk& from (6.13) and (6.15), we find the velocity of the quasi-longitudinal wave propagating in the thin-walled beam of open section
|
s |
|
G1 ¼ |
E |
ð6:16Þ |
q: |
Relationship (6.13) with due account for (6.16) takes the form
½rkk& ¼ qG1½vk&: |
ð6:17Þ |
Now multiplying (6.9) by kikj and considering that on the quasi-longitudinal wave G ¼ G1; we obtain
½vk& ¼ G1½uk; k&: |
ð6:18Þ |
Substituting (6.18) in (6.11), we are led to the following equations valid on the quasi-longitudinal wave:
½ux; x& ¼ ½uy; y& ¼ m½uk; k&: |
ð6:19Þ |
Note that in the three-dimensional medium only one value, i.e. ½uk; k&; is nonzero on the quasi-longitudinal wave, while in the one-dimensional medium, where the ‘beam-wave’ propagates, on the quasi-longitudinal wave there are two nonvanishing values, namely, ½ux; x& and ½uy; y&; resulting to the fact that, as distinct to the statics of the thin-walled beams of open section where the contour of the cross-section remains to be rigid during the process of its deformation, in the dynamic problems this contour experiences the deformation on the quasi-longitudinal wave.
It is interesting to emphasize that relationships (6.19), which are valid only on the front of the quasi-longitudinal wave propagating in the thin-walled beam of open profile, coincide by their form with the well-known formulas for simple beams of solid cross-section being under simple tension-compression, i.e.
ux; x ¼ uy; y ¼ muz; z;
which are valid for all cross sections of the simple beam with the longitudinal z-axis.
86 6 Peculiarities of Transient Wave Propagation
Multiplying (6.10) by kikj and by kisj and (6.14) by ki and by si; respectively,
we have |
|
½rkx& ¼ ½rij&kikj ¼ lG 1½vx&; ½rky& ¼ ½rij&kisj ¼ lG 1½vy&; |
ð6:20Þ |
and |
|
½rkx& ¼ qG½vx&; ½rky& ¼ qG½vy&; |
ð6:21Þ |
where ½vx& ¼ ½vi&ki and ½vy& ¼ ½vi&si.
Eliminating the values ½rkx& and ½rky& from (6.20) and (6.21), we find the
velocity of the quasi-transverse wave |
|
|
|
r |
|
G2 ¼ |
l |
ð6:22Þ |
q: |
||
Considering (6.22), relationships (6.20) take the form |
|
|
½rkx& ¼ qG2½vx&; |
½rky& ¼ qG2½vy&: |
ð6:23Þ |
Thus, we have just prove that the dynamic theory of thin-walled beams of open profile proposed in this book admits only two transient waves, quasi-longitudinal and quasi-transverse, propagating with the velocities (6.16) and (6.22), in so doing the velocity of the quasi-longitudinal wave is equal to that of longitudinal wave in a thin elastic rod of solid cross section, while the velocity of the quasi-transverse wave coincides with the velocity of the shear wave propagating in a threedimensional elastic medium.
Reference
1. T.Y. Thomas, Plastic flow and fracture in solids, (Academic Press, New York, 1961)